Communications in Contemporary Mathematics

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakke's motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen–Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) satisfies Weyl's theorem if σ(T)\σ_w(T) = π₀₀(T), where σ(T) and σ_w(T) denote the spectrum and the Weyl spectrum of T, respectively, π₀₀(T) = {λ ∈ iso σ(T) : 0 < dim N(T - λI) < ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T².

In this paper, we study the following concave–convex elliptic problems:

Let Y → ℙⁿ be a flat family of reduced Gorenstein curves, such that the compactified relative Jacobian is a Lagrangian fibration. We prove that X is a Beauville–Mukai integrable system if n = 3, 4, or 5, and the curves are irreducible and non-hyperelliptic. We also prove that X is a Beauville–Mukai system if n = 3, d is odd, and the curves are canonically positive 2-connected hyperelliptic curves.

In this paper, we study the nonexistence of positive solutions for the following elliptic equation

For any integer d ≥ 5, the Noether–Lefschetz locus, denoted NL_d, parametrizes smooth degree d surfaces in ℙ³ with Picard number at least 2. It is well-known (due to works of Voisin, Green and others) that the largest irreducible component of NL_d is of codimension (in the space of all smooth surfaces in ℙ³ of degree d) equal to d-3 and parametrizes surfaces containing a line. In this article we study for an integer 3 ≤ r < d, the sub-locus of NL_d, denoted NL_r,d, parametrizing surfaces with Picard number at least r. A conjecture of Griffiths and Harris states the largest component of NL_r,d is of codimension and the irreducible component of NL_r,d parametrizing the surfaces containing r - 1 coplanar lines is of this codimension. We prove this statement in the case r ≪ d.

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

We consider nonlinear Choquard equation

Let Mⁿ be an n-dimensional Riemannian manifold with boundary ∂M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k ∈ ℝ, we give a sharp estimate of the upper bound of ρ(x) = d(x, ∂M), in terms of the mean curvature bound of the boundary. When ∂M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kähler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kähler manifold and also estimate the first eigenvalue of the real Laplacian.

We give a family of monotone quantities along smooth solutions to the inverse curvature flows in Euclidean spaces. We also derive a related geometric inequality for closed hypersurfaces with positive kth mean curvature.

Let U(n) be the quantum enveloping algebra of 𝔤𝔩_n over ℚ(v). We will use q-Schur algebras to realize the Lusztig integral form of U(n). Furthermore, we will use this result to realize quantum 𝔤𝔩_n over k, where k is a field containing an lth primitive root ε of 1 with l ≥ 1 odd.

We introduce a pair of middle Cantor sets which have stable intersection, while the product of their thickness is smaller than one.

The aim of this paper is to employ a strategy known from fluid dynamics in order to provide results for the linear heat equation u_t - Δu - V(x)u = 0 in ℝⁿ with singular potentials. We show well-posedness of solutions, without using Hardy inequality, in a framework based in the Fourier transform, namely, PM^k-spaces. For arbitrary data u₀ ∈ PM^k, the approach allows to compute an explicit smallness condition on V for global existence in the case of V with finitely many inverse square singularities. As a consequence, well-posedness of solutions is obtained for the case of the monopolar potential with . This threshold value is the same one obtained for the global well-posedness of L²-solutions by means of Hardy inequalities and energy estimates. Since there is no any inclusion relation between L² and PM^k, our results indicate that λ_* is intrinsic of the PDE and independent of a particular approach. We also analyze the long-time behavior of solutions and show there are infinitely many possible asymptotics characterized by the cells of a disjoint partition of the initial data class PM^k.

Let be the Lie algebra of Block type with basis {L_i,j | i,j ∈ ℤ} and relations [L_i,j, L_k,l] = ((j + 1)k - (l + 1)i)L_i+k,j+l. Since is ℤ²-graded, it is natural to study ℤ²-graded modules over . In this paper, ℤ²-graded -modules of the intermediate series are classified.